optimization of an electrodynamic loudspeaker, the analysis of an ... Therewith, the whole structure, consisting of the coil, former, suspension, surround and cone ...
Large Scale Computation of Coupled Electro-Acoustic Systems using ANSYS and CAPA H.Landes, M. Kaltenbacher, R. Lerch
Chair of Sensor Technology, University of Erlangen-Nürnberg
Summary: The numerical simulation of coupled electro-acoustic problems requires the precise and efficient computation of electric, magnetic, mechanic, and acoustic fields including their mutual couplings. To meet these requirements, the finite-element-boundary-element program CAPA has been developed by the authors during the last years. Recently we have implemented an interface between CAPA and ANSYS, which allows the pre- and postprocessing tasks to be performed within ANSYS while the numerical calculations are carried out in CAPA. Therewith, we have successfully combined the user-friendly interface of ANSYS with the dedicated computational capabilities of CAPA. Several examples demonstrate the applicability of this approach: the optimization of an electrodynamic loudspeaker, the analysis of an electromagnetic acoustic transducer (EMAT) as used in nondestructive testing, and the simulation of a controlled micromachined ultrasound array.
Keywords: Finite elements, boundary elements, coupled field simulations, multiphysics, large scale computations
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1. Introduction From a modern point of view, the development of electro-acoustic devices asks for powerful computer-aidedengineering (CAE) tools. In recent years, numerical methods have been used extensively in the field of engineering design. So Finite-Element-Methods (FEM) and Boundary-Element-Methods (BEM) are well established in the area of mechanical and electrical engineering [1,2,3]. With these computer simulations, the costly and lengthy fabrication of a large number of prototypes, required in optimization studies by conventional experimental design, can be reduced tremendously. As a representative example of an electro-acoustic actuator consider an electrodynamic loudspeaker, as shown in Fig. 1. A cylindrical, small light voice coil is suspended freely in a strong radial magnetic field, generated by a permanent magnet. When the coil is loaded by an electric voltage signal, the interaction between the magnetic field of the permanent magnet and the current in the voice coil results in an axial Lorentz force, acting on the coil. Therewith, the whole structure, consisting of the coil, former, suspension, surround and cone diaphragm, starts to move and generates the acoustic sound. Since in the case of a electrodynamic loudspeaker the interaction with the ambient fluid must not be neglected, the electrodynamic loudspeaker represents a typical coupled magnetomechanical system immersed in an acoustic fluid. That is the reason, why for the detailed finite element modeling of these moving coil drivers the magnetic, the mechanical as well as the acoustic fields including their couplings have to be considered as one system, which cannot be separated. Due to the complexity of these multi-field problems and the fact, Fig. 1: Schematic of an electrodynamic loudspeaker that for an efficient simulation computations have to be performed in time domain by means of a transient analysis, the straight forward application of standard simulation tools has shown only limited success.
2. Theory For the computer simulation of electromagnetic actuators immersed in an acoustic fluid, such as an electrodynamic loudspeaker, the physical fields and couplings, as shown in Fig. 2 have to be modeled.
Fig. 2: Considered fields and their couplings of an electrodynamic loudspeaker NAFEMS-Seminar: ’’Computational Acoustics’’ – November 10-11, 1999 -2-
In the case of a coupled electrostatic-mechanical-acoustic problem, the magnetic field equation in Fig. 2 has to be replaced by the electric potential equation, which is given by The coupling between electrostatics and mechanics is now given by the electrostatic force, which may be calculated using the electrostatic force tensor TE,
Herein, Ex, Ey, and Ez denote the x-, y-, and z-components of the electric field E, respectively. Therewith, the electrostatic force is computed as
with n the normal vector of the surface A. In this paper, the equations governing the electromagnetic, mechanical, and acoustic field quantities are solved using a Finite-Element-Method (FEM) by means of the simulation software CAPA [4]. The theory of the underlying equations and finite element scheme have already been reported in [5,6,10] and will not be repeated here.
3. ANSYS-CAPA Interface Due to the complexity of the considered electro-acoustic devices, tools are required, which allow for an userfriendly and effective modeling. Furthermore, since in these simulations the radiation and the propagation of an acoustic wave in a large or even unbounded fluid medium has to be considered, typically very large finite element meshes result. Since the ANSYS software offers fast and reliable meshers as well as an user-friendly interface, we have decided to meet the needs of our modeling requirements by implementing an ANSYS-CAPA interface. Therewith, all pre- and postprocessing can now be performed within ANSYS, while the simulation itself is run within CAPA. The interface itself is written in C++, therewith, is portable between different computer platforms. It has already been tested on various UNIX workstations, as well as PCs running under LINUX and WINDOWS NT. In the implementation of the interface, the user programable features of ANSYS have been used [7]. All interactions with the interface are realized through the USER01 subroutine of ANSYS and all CAPA elements are mapped onto the ANSYS elements UEC100, UEC101, and UEC102. Furthermore, since a complete coverage of the CAPA features by means of ANSYS commands could not be realized in a reasonable manner, which was true especially for the definition of a transient analysis, it was decided to extend the ANSYS command set by more appropriate commands. Therewith, the definition of a CAPA analysis is made as easy as an internal ANSYS analysis.
4. Electrodynamic loudspeaker
Fig. 3: Finite element model of electrodynamic loudspeaker
In case of an electrodynamic loudspeaker the voice coil and aluminum former are discretized by so-called magnetomechanical finite elements, which solve the equations governing the magnetic and mechanical field quantities and take account of the full coupling between these fields (see Fig. 2). Due to the concentration of the magnetic flux within the magnet assembly, the magnet structure and only a small ambient region have to be discretized by magnetic finite elements. Furtheron, the surround,
NAFEMS-Seminar: ’’Computational Acoustics’’ – November 10-11, 1999 -3-
suspension, dust cap and cone diaphragm are modeled by standard mechanical finite elements. Finally, the fluid region in front of the loudspeaker is discretized by acoustic finite and infinite elements. In order to function properly, the infinite elements have to be located in the far field of the moving coil driver. Consequently, a large number of acoustic finite elements is necessary in the modeling of electrodynamic loudspeakers. 4.1 Verification of the computer model The verification of both computer models described above has been performed by comparing simulation results with corresponding measured data. Therefore, the frequency dependency of the electrical input impedance as well as the axial pressure response of the electrodynamic loudspeaker have been calculated. For the pressure calculation and measurement, the input source is a voltage with nominal 1 W referred to 8 Ω (2.83 V r.m.s.). The microphone position is on the mid axis at a distance of 1 m from the loudspeaker. As can be seen in Fig. 4, good agreement between both simulation results and measured data was achieved.
Fig. 4: Comparison of simulation and measurment 4.2 Investigation in Design Parameters As a first application of our calculation scheme in the computer-aided-design of electrodynamic loudspeakers, the elimination of response dips at intermediate frequencies was considered. Measurements as well as simulation results reveal two dips in the sound pressure response occuring at approx. 400 Hz and 900 Hz (see Fig. 4). The elimination of these dips is of great interest for loudspeaker manufacturers, since a flat axial pressure response over a wide frequency range is desired [8].
Fig. 5: Design improvements for electrodynamic loudspeaker NAFEMS-Seminar: ’’Computational Acoustics’’ – November 10-11, 1999 -4-
In computer simulations two design modifications could be established, both leading to the elimination of these response dips. In Fig. 5 the comparison of the axial pressure response with original and modified designs is presented. As can be seen, increasing the loss-factor by a factor of 2.5 results in a more effective absorption and termination of the outward travelling energy and in reduced response dips. In the case of the modified surround, i.e. when a flat section is added in the surround, the change of surround mass and compliance results in a modified equivalent circuit and causes the elimination of these response dips [8]. With both modifications the deviation can be held within 1 dB over a wide frequency range and, therefore, an improvement in respect to response flatness can be achieved.
5. Electromagnetic acoustic transducer Fig. 6 shows a typical setup of an electromagnetic acoustic transducer (EMAT) as used for generation and reception of plate waves in thin metallic sheets. The permanent magnet subjects the area under the coil to a static magnetic field oriented mainly perpendicular to the surface of the sheet. For efficient generation of Lamb waves the wirespacing of the coil is typically chosen to be larger than the plate-thickness. The coil, loaded with an alternating current, produces a time varying magnetic field, which in turn induces eddy currents in the material under test. The interaction of these eddy currents with the overall magnetic field of the permanent magnet and the coil results in a distribution of the magnetic volume force (Lorentz force) showing a spatial periodicity, which is equal to the double wire Fig. 6: Setup of a plate wave EMAT spacing of the meander coil. Therewith, a plate wave is generated and propagates along the sheet. Using an EMAT as a receiver for ultrasonic waves, the setup is essentially the same as for the transmitting EMAT. When the plate wave passes the region of the receiving EMAT, which is subjected to the static magnetic field of the permanent magnet, locally eddy currents are induced in the conductive metallic sheet. Therewith, the time varying magnetic field of these eddy currents induces the voltage in the meander coil. A detail of the finite element model which has been generated using ANSYS is shown in Fig. 7 whereas some aspects of the calculated magnetic field are depicted in Fig. 8.
Fig. 7: 3D finite element model of an EMAT
Fig. 8: Magnetic field of an EMAT
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The resulting Lorentz forces, which lead to the exciatation of the plate wave in the material under test, are shown in Fig. 9. This numerical simulation scheme for EMATs has already been applied in the design optimzation of electromagnetic acoustics transducers [9].
Fig. 9: Lorentz forces generated by an EMAT in metalic plate
6. Micromachined capacitive ultrasound array As an application of our numerical calculation scheme, an array consisting of 19 capacitive transducer cells, as shown in Fig. 10, was considered [11]. In Fig. 11, the finite element model of a single array is displayed. The membranes had a thickness of 1 µm and the gap between the electrodes was 500 nm. A DC voltage of 10 V was applied to the electrodes and a single period of a sine burst with frequency 5 MHz and amplitude 10 V was used for excitation. During experiments a very long ring down time of the membranes had to be noticed. Furthermore, due to the fluid-solid coupling, a strong crosstalk between the individual membranes was detected. Therefore, investigations focused on these topics have been performed.
Fig. 10: Topview of a CMOS chip with 4 arrays, each containing 19 capacitive transducers
Fig. 11: Finite element model of a single transducer array
In order to decrease the ring down time of the membranes, our finite element model was expanded by an external controller. Due to the quadratic dependency of the electrostatic force on the deflection of the membranes, a nonlinear controller has been designed. The change of the capacitance of each transducer is computed from the mechanical displacements and used as the input of the controller. The controller algorithm then calculates the voltage for each transducer, which is a direct input value for the electric source. Using this nonlinear controller, the secondary signal in the acoustic pressure, as observed for the uncontrolled case, is no longer present for the controlled membrane array. This is shown in Fig. 12 for the case, that all membranes are driven in parallel. As a consequence a smoothing effect of the controller is also observed in the frequency spectrum (Fig. 13).
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500
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pressure (Pa)
300 200 100 0 -100
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Fig. 12: Pressure signal of controlled and uncontrolled array
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Fig. 13: Frequency spectrum of controlled and uncontrolled array
7. Conclusion The application of a combination of ANSYS and CAPA to some questions in the design of electro-acoustic transducers has been reported. With the availability of the ANSYS-CAPA interface, user-friendly modeling is brought together with the dedicated computational capabilities of CAPA, and, therewith, design optimization of such transducers by numerical methods is made much more easier.
8. References [1] R. Lerch, M. Kaltenbacher, H. Landes, F. Lindinger, Computerunterstützte Entwicklung elektromechanischer Transducer, e&i ÖVE-Verbandszeitschrift, Vol.7/8, pp. 532-546, 1996 [2] O.C. Zienkiewics, R.L. Taylor, The Finite Element Method, McGraw-Hill Book Company, London, 1991 [3] C.A. Brebbia, J. Dominguez, Boundary Elements-An Introductory Course, Computational Mechanics Publications, Southampton, 1992 [4] H. Landes, R. Lerch, M. Kaltenbacher, CAPA User Manual, Vol. 3.2, 1999, University of Linz, Altenberger Str. 69, A-4040 Linz [5] M. Kaltenbacher, H. Landes, R. Lerch, An Efficient Calculation Scheme for the Numerical Simulation of Coupled Magnetomechanical Systems, IEEE Trans. on Magnetics, Vol. 33, No. 2, March 1997 [6] M. Kaltenbacher, M. Rausch, H. Landes, R. Lerch, Numerical modelling of electrodynamic loudspeakers,
COMPEL, Volume 18, Issue 3 (to be published) [7] Guide to ANSYS User Programmable Features, Release 5.5, ANSYS, Inc., 1998 [8] M. Colloms, High Performance Loudspeakers, John Wiley & Sons, Chichester, New York, 1997 [9] M. Kaltenbacher, R. Lerch, H. Landes, K. Ettinger, B. Tittmann, Computer Optimization of Electromagnetic Acoustic Transducers, IEEE International Ultrasonics Symposium, Japan, 5.-8.October 1998 [10] M. Kaltenbacher, H. Landes, K. Niederer, R. Lerch, 3D Simulation of Controlled Micromachined Ultrasound Transducers, IEEE International Ultrasonics Symposium, Lake Tahoe, 18.-20.October 1999 [11] C. Eccardt, K. Niederer. T. Scheiter, C. Hierold., Surface micromachined ultrasound transducers in CMOS technology, Proc. Ultrasonics Symposium, San Antonio, vol. 2 , pp. 959-962, 1996
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